Determination and also make the joints work

Determination of the strain energy
function using stress-strain response of
a single fascicle for the modeling of ligaments and tendons

Md Asif Arefeen

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review and analysis of the strain energy function by using the distribution of crimp angles of the fibrils to
determine the stress-strain response of single fascicle. (Kastelic, Palley et al. 1980) gave a
non-linear stress-strain relationship based on the radial variation of the
fibril crimp. By correcting this relationship Tom
Shearer derived a new strain energy function and compared it with the commonly
used model HGO. The relative and absolute errors related to the new model are
less than 9% and 40% than of that HGO model. Undoubtedly
new model gives a better performance than the HGO model. But
is mandatory to measure the


o separately for
the ligament or tendon in order to validate this model.


1. Introduction

A fascicle
is the main subunit of the ligaments and tendons which are the soft collagenous
tissue. These tissues are the fundamental
structures of in the musculoskeletal systems and play a significant role in biomechanics.
Ligaments provide stability and also make the joints work perfectly by
connecting bone to bone, on the other hand, tendons transfer force to a skeleton which is generated by muscle by
connecting bone to muscle. The collagenous fibers
like fascicle consist of crimped pattern
fibrils and this crimp are called the waviness
of the fibrils(see fig.1) which contributes significantly to the non-linear
stress-strain response for ligaments and tendons.As an anisotropic tissue, the
characteristic of stress-strain of ligaments and tendons within a non-linear
elastic framework occur in the toe region where mechanically loading of the
tendon up to 2% strain(see fig.2).




                         Fig 1. Tendon hierarchy                                               
Fig 2. Model within a non-linear framework



(Fung 1967) gave an
exponential stress-strain relationship based on rabbit
mesentery which was only in a phenomenological
sense but there was no microstructural basis for the choice of the
exponential function. Based on his work (Gou 1970) proposed a strain energy function for isotropic tissues
that also gave an exponential stress-strain relationship but was not suitable
for tissue like tendons and ligaments. (Kastelic, Palley et al. 1980) gave a
non-linear stress-strain relationship based on the radial variation of the fibril crimp. But there was an error in the
implementation of the Hook’s law which leads
his relationship incorrect. The strain energy function which has used for modeling biological tissue for a
long time is Holzapfel-grasser-Ogden (HGO) model,
given by


(I1-3) +


-1), where, I1= trC, I4=
M.(CM),  C=

and I4 are the strain invariants where I4 has a
direct interpretation as the square of the stretch in the direction of the fiber.More explanations about invariants can be found
in the (Holzapfel et al. 2010).”C is the right Cauchy-Green tensor, F is the
deformation gradient tensor and M is a unit vector pointing in the direction of
the tissue’s fibers before any deformation has taken place, c, k1and
k1 are material parameters and the above expression is only valid
when I4?1(when I4>1,
W =

(I1-3)). As a
phenomenological model, the parameters are not directly linked to measurable
quantities”.So this model has some limitations.

A large number SEF model has been proposed so far by
different researchers like( Humphrey and Lin 1987),(
Humphrey et al.1990), (Fung et al. 1993),( Taber 2004), (Murphy
2013) but none of them were valid for ligaments and tendons.In 2014 Tom Shearer
proposed a model by correcting the work done by Kastelic based on the fibril
crimp angle.This new model is more efficient than the HGO model.


2.Development of new
stress-strain relationship

A new stress-strain response has given by the Tom Shearer
based on the radial variation in the crimp angle of a fascicle’s fibrils by
correcting the Hook’s law in that paper.The Hook’s law stated by Kastelic et
al.(1980) is given by

?p(?)=E*. ??p (?), where ??p (?)= ? – ?p (?)

Here ??p (?)(elastic-deformation)
is not the fibril strain and differs from the fibril strain by a quantity that is
dependent on ?.All fibrils should have same Young’s modulus.So E* is not valid
for all ?.New Hook’s law was given by Tom Shearer in his paper which
can be derived from the figure-3 below.






( ? – ?p (?))= ( ? +1) cos(

-1= ( ? +1) cos(



Fig 3: Stretching of fibril of initial length lp(?)
within a fascicle of initial length L


Using the equation (1) he derived an expression for the
average traction in the direction of the fascicle

= 2

Pp is the tensile load faced by the fascicle. Taking p=1,2 and
simplifying few things Tom Shearer derived a new stress-strain relationship which is given by





( ? +1)-1,                      ?=

= E(??-1),                           ?>


Shearer used this form to derive the new strain energy function.

3. Strain Energy Function

this section, a derived strain energy function will be shown for the ligaments
and tendons. For the details, the reader
is referred to Tom Shearer (2014).His strain energy function is valid for both of
the isotropic and anisotropic tissue.

anisotropic tissue SEF



I4 -3log (I4)-


neo-Hookean model is still reasonable for isotropic
tissue”. Based on this an isotropic SEF
can be derived



full form of strain energy function can be given as


(I2-3) +


I4 -3log(I4)-




(I2-3) +  


I4 –

log(I4)+?),                  I4



the collagen volume fraction, E is the
fibril stiffness and

 is the average out fibril crimp angle. Here

 cannot be measured directly. As a result, it was taken based on assumptions.
Finally, the above SEF gives stress-strain response for both isotropic and anisotropic tissues. It seems
quite unusual for isotropic SEF but it happens due to the inability of the linear term in their stress-strain relationship
for small strains of fascicles.

4. Result

this section, a comparison of the stress-strain relationship among new model, HGO model, an
experimental model will be shown. The existing data were taken from the (Johnson, Tramaglini et al. 1994), Parameter values: c=(1-?)

=0.01MPa, k1=25MPa, k2=183MPa,

=552 MPa,

=0.19 rad=10.7?.As stiffness of ligament
and tendon matrix is insignificant compared with that of its fascicles, (1-?)

 were chosen to be small,

 cannot be measured directly , it was taken based on assumptions like 0.11

1. Also

 was not available so it was taken as a
predicted value. Based on this Tom Shearer measured the stress-strain response
which is given below

4: Comparison stress-strain curves of
the new model and HGO model with
experimental data. Black: new model, Blue: HGO model, Red: experimental data.


From the above graph,
an average relative error and absolute
error among the model can be calculated.
Calculation of the Tom Shearer suggested that average relative error and
absolute error of new model is less than the HGO model respectively 0.051 (new
model)<0.59(HGO) and 0.13MPa (new) < 0.35 MPa (HGO).   5. Conclusion Undoubtedly new model gives a better performance than the HGO model. But after reviewing and analyzing different kinds of literature it is mandatory to measure the and o separately for the ligament or tendon in order to validate this model.   6. Reference Fung, Y. C. (1967). "Elasticity of soft tissues in simple elongation." Am J Physiol 213(6): 1532-1544.                 Gou, P. F. (1970). "Strain energy function for biological tissues." J Biomech 3(6): 547-550.                 Johnson, G. A., et al. (1994). "Tensile and viscoelastic properties of human patellar tendon." J Orthop Res 12(6): 796-803.   Kastelic, J., et al. (1980). "A structural mechanical model for tendon crimping." J Biomech 13(10): 887-893. Johnson, G.A., Rajagopal, K.R., Woo, S.L-Y.,1992."A single integral finite strain(SIFS) model of ligaments and tendons".Adv.Bioeng.22,245–248. Holzapfel, G.A., Gasser, T.C., Ogden, R.W.,2000."A new constitutive framework for arterial wall mechanics and a comparative study of material models". J.Elast.61,1–48 Shearer, T., Rawson, S., Castro, S.J., Ballint, R., Bradley, R.S., Lowe, T., Vila-ComamalaJ., Lee, P.D., Cartmell, S.H., 2014." X-ray computed tomography of the anterior cruciate ligament and patellar tendon. Muscles Ligaments Tendons". J.4, 238–244.